A 40 g marble moving at 2.3 m/s strikes a 29 g marble at rest. Assume the collision is perfectly elastic and the marbles collide head-on.
What is the speed of the first marble immediately after the collision?
What is the speed of the second marble immediately after the collision?
I've tried multiple ways of setting this up and each time its not right
Why was this question given 5 thumbs down?
Well, looks like we have a marble-on-marble collision here. Let's see if we can unravel this puzzle using a pinch of physics and a dash of laughter.
To find the speed of the first marble after the collision, we need to apply the law of conservation of momentum. When the marbles collide, the total momentum before the collision should be equal to the total momentum after the collision. Don't worry, I won't go off on a momentum monologue—I'll keep it light!
Now, let's calculate the initial momentum of the marbles before the collision. Momentum is given by the formula: momentum = mass × velocity. So, the initial momentum of the first marble would be 40 grams (or 0.04 kilograms) × 2.3 m/s, which equals... spotlight, please... 0.092 kg·m/s. Ta-da!
Since we're assuming a perfectly elastic collision, momentum is conserved. So, both marbles should share the total momentum equally. Dividing the initial momentum (0.092 kg·m/s) by the total mass of both marbles (40 g + 29 g = 69 g = 0.069 kg), we get the speed of the first marble after the collision. Drumroll, please... approximately 1.34 m/s. Yay, the first marble made it through!
But wait, we haven't forgotten about the second marble. To find its speed after the collision, we can use the same formula for momentum. The momentum of the second marble before the collision would be 29 grams (or 0.029 kilograms) × 0 m/s (since it's at rest), which equals... a surprising zero. Sorry, second marble, you didn't have any momentum to start with.
After the collision, the second marble will gain momentum since momentum is conserved. Dividing the total momentum (0.092 kg·m/s) by the total mass (0.069 kg) again, we find the speed of the second marble after the collision. And the answer is... ta-da! 1.34 m/s. Both marbles end up with the same speed. How sweet, they're matching!
So, the speed of the first marble immediately after the collision is approximately 1.34 m/s, and the speed of the second marble immediately after the collision is also approximately 1.34 m/s. It's a tie! The marbles survived the collision, and physics didn't get the last laugh this time. Keep rolling and stay elastic, my marble friends!
To solve this problem, we can use the principle of conservation of momentum and apply it to each marble separately. The conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision.
Step 1: Find the initial momentum of each marble:
Momentum (p) is defined as the product of mass (m) and velocity (v).
For the first marble:
Mass (m1) = 40 g = 0.04 kg
Velocity (v1) = 2.3 m/s
Initial momentum of the first marble (p1_initial) = m1 * v1
For the second marble:
Mass (m2) = 29 g = 0.029 kg
Velocity (v2_initial) = 0 m/s (since it is at rest)
Initial momentum of the second marble (p2_initial) = m2 * v2_initial
Step 2: Calculate the total momentum before the collision:
Total initial momentum (p_initial) = p1_initial + p2_initial
Step 3: Calculate the total momentum after the collision using the principle of conservation of momentum:
Total final momentum (p_final) = p1_final + p2_final
Since it is given that the collision is perfectly elastic, the total momentum before and after the collision should be equal.
Step 4: Set up equations using the conservation of momentum:
p_initial = p_final
Step 5: Solve for the final velocities of the marbles:
For the first marble:
Final momentum of the first marble (p1_final) = m1 * v1_final
For the second marble:
Final momentum of the second marble (p2_final) = m2 * v2_final
Using the equation p_initial = p_final, we can set up the following equations:
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
Step 6: Solve the equations to find the final velocities:
Substitute the given values into the equation and solve for v1_final and v2_final.
(0.04 kg * 2.3 m/s) + (0.029 kg * 0 m/s) = (0.04 kg * v1_final) + (0.029 kg * v2_final)
Since the second marble is initially at rest, its initial velocity (v2_initial) is 0 m/s.
(0.092 kg * m/s) = (0.04 kg * v1_final) + (0.029 kg * v2_final)
So, the equation now becomes:
0.092 kg * m/s = 0.04 kg * v1_final + 0.029 kg * v2_final
Step 7: Solve the equation for v1_final and v2_final:
By rearranging the equation, we can find the values of v1_final and v2_final.
0.04 kg * v1_final = 0.092 kg * m/s - 0.029 kg * v2_final
v1_final = (0.092 kg * m/s - 0.029 kg * v2_final) / 0.04 kg
Step 8: Finally, substitute the value of v1_final into the equation to find v2_final:
By substituting the calculated value of v1_final into the equation, we can solve for v2_final.
0.092 kg * m/s = 0.04 kg * ((0.092 kg * m/s - 0.029 kg * v2_final) / 0.04 kg) + 0.029 kg * v2_final
Simplifying the equation:
0.092 kg * m/s = 0.092 kg * m/s - 0.029 kg * v2_final + 0.029 kg * v2_final
0.029 kg * v2_final = 0
Therefore, we can see that v2_final = 0 m/s.
Step 9: Calculate v1_final:
Substituting the value of v2_final = 0 into the equation for v1_final:
v1_final = (0.092 kg * m/s - 0.029 kg * 0) / 0.04 kg
v1_final = (0.092 kg * m/s) / 0.04 kg
Therefore, the speed of the first marble immediately after the collision is 2.3 m/s, since the velocity remains the same.
Step 10: Calculate v2_final:
From Step 8, we found that v2_final = 0 m/s.
Therefore, the speed of the second marble immediately after the collision is 0 m/s.
To determine the speeds of the marbles immediately after the collision, we can use the law of conservation of momentum and the law of conservation of kinetic energy. Let's calculate the speeds step by step.
Step 1: Calculate the initial momentum of the system
The initial momentum of the system is equal to the sum of the individual momentums of the marbles before the collision.
The momentum (p) of an object is given by the product of its mass (m) and velocity (v): p = mv.
For the first marble:
momentum1 = mass1 * velocity1
= 40 g * 2.3 m/s
For the second marble at rest:
momentum2 = mass2 * velocity2
= 29 g * 0 m/s
= 0
Step 2: Apply the law of conservation of momentum
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
momentum1_initial + momentum2_initial = momentum1_final + momentum2_final
(40 g * 2.3 m/s) + (29 g * 0 m/s) = (40 g * v1) + (29 g * v2)
Step 3: Apply the law of conservation of kinetic energy
In a perfectly elastic collision, both momentum and kinetic energy are conserved. The kinetic energy (KE) of an object is given by the formula: KE = (1/2)mv^2.
Applying the law of conservation of kinetic energy:
(1/2) * (mass1 * velocity1)^2 + (1/2) * (mass2 * velocity2)^2 = (1/2) * (mass1 * v1)^2 + (1/2) * (mass2 * v2)^2
We can now solve these equations to find the speeds after the collision.
Let's convert the masses to kilograms to work in the SI unit system:
mass1 = 40 g = 0.04 kg
mass2 = 29 g = 0.029 kg
Solving these equations will give us the final velocities (v1 and v2).
Please note that I will now calculate the final velocities using the given data.
The center of mass (CM) moves at a velocity
Vcm = 0.40*2.3/0.69 = 1.333 m/s
In a coordinate system moving with the center of mass, the 40g marble moves at speed
2.3 - 1.333 = 0.967 m/s before collision and -0.967 m/s after collision. In that same coordinate system, the 29 g marble moves at -1.333 m/s before collision and +1.333 m/s after collision.
Now transfer final velocities in CM coordinates back to laboratory coordinates by adding 1.333 m/s to each.
The 40g marble moves at 0.366 m/s and the 29 g marble moves at 2.666 m/s after collision. (The latter is the answer you wanted)
Momentum check (in lab coordinates):
Initial momentum = 0.40*2.3 = 0.920 kg m/s
Final momentum = 0.40*0.366 + 0.29*2.666 = 0.920 m/s
Kinetic energy is also conserved.